Quantification of dissipation and deformation in ambient atomic force microscopy

A formalism to extract and quantify unknown quantities such as sample deformation, the viscosity of the sample and surface energy hysteresis in amplitude modulation atomic force microscopy is presented. Recovering the unknowns only requires the cantilever to be accurately calibrated and the dissipative processes occurring during sample deformation to be well modeled. The theory is validated by comparison with numerical simulations and shown to be able to provide, in principle, values of sample deformation with picometer resolution.


2
where A 0 is the free amplitude, k is the spring constant and Q is the Q factor due to dissipation with the medium. While, in ambient conditions, higher harmonics have been shown to reduce to a few angstroms or fractions of angstroms of amplitude, these can still be accounted for in order to increase the accuracy of (1). According to Tamayo [9], where A n are the amplitudes of the harmonics (A = A 1 in (1)). Further accuracy in E dis could be obtained by considering the second mode of oscillation [10,11]. Other advances in the in situ calibration of k, Q, the natural oscillation frequency ω 0 and the tip radius R have been made in parallel and we consider these as known experimental quantities in the rest of this work [12][13][14].
For simplicity, we further set ω = ω 0 throughout. With all, E dis in (1) and (2) is a quantity that accounts for the net energy dissipated but the expression cannot distinguish between dissipative processes. Thus, effort has been put into interpreting and/or decoupling the different contributions to E dis over the last few years. For example, Garcia et al [1,2,15] found that it is dE dis /dA that provides information about the nature of the dissipative mechanisms. In particular, during sample deformation, two main dissipative processes have been identified, namely surface energy γ hysteresis and viscoelasticity. It was also proposed that in the repulsive force regime and by sufficiently increasing the value of the free amplitude A 0 , the dissipation and processes occurring during sample deformation could be probed; both sample deformation and dissipation scale with the free amplitude [1,16,17]. Also note that prevailing long-range dissipative processes, such as the hysteretic formation and rupture of the capillary neck that might occur in ambient conditions, should be relatively independent of operational parameters and can be readily probed in the attractive force regime [3,18]. More recently, viscoelastic heterogeneity in a metallic glass has been mapped with nanoscale resolution using the dE dis /dA approach. Such experiments provide fundamental experimental information that might be used to fill the gap between atomic models and macroscopic glass properties [8]. Moreover, due to the good agreement between the dE dis /dA method and experiment [1,2,19], several other approaches to identifying and decoupling dissipative processes [20,21], some of these being quantitative [21], have recently been proposed. Nevertheless, quantitative approaches might have disadvantages such as the requirement to select an appropriate model for the conservative component of the force or other dissipative channels, in particular in the long range where mechanical contact does not occur [21,22]. Garcia et al further found that viscoelastic and surface energy hysteresis processes could be well modeled by considering (i) the variation in the adhesion force during tip approach F AD (approach) = −4 Rγ a and tip retraction F AD (retraction) = −4 Rγ r , where the respective surface energies are γ a and γ r (γ r γ a ) and (ii) the Voigt model for the viscoelastic force F η [2,19,23], where δ is the instantaneous tip-sample deformation,δ is the tip velocity during deformation and η is the viscosity of the materials in contact. Note that tip-sample deformation here refers to the fact that when the tip and the sample are in mechanical contact, both materials deform [24]. If the elastic modulus of the tip is much greater than that of the sample the deformation can be associated with the sample alone. Nevertheless, in this work, the terms sample deformation and 3 tip-sample deformation will be used interchangeably to refer to tip-sample deformation and will be written as δ. The dissipative term of the adhesion force F α can also be written as where α = γ r /γ a − 1 accounts for differences in γ a and γ r and where γ = γ a and F AD = −4 Rγ . More recent studies also support the validity of these expressions for a variety of samples where large values of α have been interpreted as the capacity to restructure the atomic bonds of the surfaces in contact [7,21,25]. It is clear that the interpretation of α and η requires an understanding of dissipation in a dynamic nanoscale contact. However, here, we assume that the two main dissipative channels during sample deformation are surface energy γ hysteresis and viscoelasticity. In particular, we further assume that these two processes are well modeled with (3) and (4), respectively. This is done as a first approximation to the problem with the understanding that future experimentation and testing should establish whether these expressions are valid in general or whether slight modifications will be required. Then, we set to provide the fundamental relationships between the known dynamic (static) parameters, i.e. A, A 0 , ω and z 0, where z 0 is the mean cantilever deflection, Q, and ϕ (k and R) and what are termed, from now on, the sample's dissipative properties, that is, α and η. The formalism developed in this work further provides the means to quantify the maximum tip-sample deformation in one cycle δ M , this typically being another fundamental unknown in experiments. Our simulations show that, with the proposed formalism, δ M can be potentially quantified with great accuracy and down to picometer resolution with the use of known experimental parameters. In particular, the framework depends on the relationships between the magnitude of the energy dissipated in one cycle E dis , that is, (1) or (2), and the dependencies of the dissipative channels, i.e. viscoelasticity and surface energy hysteresis, on δ M and on the oscillation amplitude A. Integration of (3) and (4) shows that the energy dissipated in one cycle via surface energy hysteresis E α depends linearly on δ M ,whereas the dependence is quadratic for the energy dissipated via viscoelasticity E η . A great advantage of the present approach is that it is independent of the nature of the tip-sample interaction provided the two main dissipative processes during sample deformation are surface energy hysteresis and/or viscoelasticity. Experimentally, one could first use the dE dis /dA method [1,8,20] to establish whether these dissipative processes are dominant and then quantify the dissipative parameters and the tip-sample deformation with the present formalism. It should be noted, however, that only the energy dissipated during sample deformation should be decoupled from the energy dissipated during non-contact dissipative processes. That is, care should be taken when establishing whether the energy is being dissipated during sample deformation alone. In this respect, several methodologies are proposed at the end of this work. One of the proposed methodologies is used satisfactorily in a numerical example. In summary, this work provides a framework to directly obtain quantitative information about fundamental dissipative properties of the sample with the use of known experimental parameters alone. Let us begin by writing the relationships between δ M and the energy dissipated in the two processes (3) and (4). It follows that where, as stated, E α is the energy dissipated due to surface adhesion hysteresis in one cycle. In order to express the energy dissipated per cycle via viscoelasticity E η as a function of δ M use is where d is the instantaneous tip-sample distance, z c is the cantilever-sample separation and z is the instantaneous position of the tip relative to the cantilever. The geometrical relationships between these parameters are shown in figure 1. Experimentally z can be expressed as a Fourier series to any degree of accuracy, i.e. z = z 0 + n=∞ n=1 A n cos(ω n t + φ n ). Also note that while both d and z c are unknown in absolute terms, differences in z c for two data points, i.e. z c , are known experimentally to the accuracy of the AFM z-piezo. This fact will be employed later to extract δ M , α and η from the known parameters. Now, from z, only the minimum value, i.e. z m in figure 1, is required in order to relate d to z, where d m is the minimum distance of approach in the cycle. During sample deformation there is no loss of generality (see figure 1) in writing z as a function of δ and δ M , While the exact value of z min can be known experimentally from the Fourier series, an approximation is used here to obtain a closed form for E η . Let us assume that the cantilever is sufficiently stiff that higher harmonics can be neglected. Then, z min ≈ z 0 − A and [7] To the same degree of accuracy and writing = ( Finally, from (10) and noting that F η is an even function of δ or more compactly where β is a function of known parameters (β = √ 2/4π R 1/2 ω). Provided viscoelasticity (3) and surface adhesion hysteresis (4) are the two dissipative processes in the mechanical contact, (5) and (11)-(12) fully account for the total energy dissipated during sample deformation. If the instantaneous contact radius r is now taken from a contact mechanics model, several interesting parameters can be computed. For example, from the Derjaguin-Muller-Toporov (DMT) model [26] it follows that r = (Rδ) 1/2 . Then, the maximum contact area S M during sample deformation is S M = Rδ M . The net values of areal energy density ρ[eVnm −2 ] and localization of energy M [eV nm −4 ] [27] as well as the particular components due to F α and F η , i.e. ρ α , M α , ρ η and M η , can be computed. For the latter It has recently been reported, however, that both ρ and M can be underestimated by using S M since the energy is not evenly dissipated throughout S M during each cycle [7,27]. Thus, an effective area S α can be defined as where T is the period of oscillation. Similarly, S η = 3/5S M . Then, ρ α = 3/2ρ α , M α = 9/4ρ α , ρ η = 5/3ρ η and M η = 25/9M η . It remains to be shown that α, η and δ M can be found with the use of known experimental parameters. In the remainder of this work, the energy conservation principle is used to extract α, η and δ M with the use of the known experimental parameters alone. The two main assumptions are that (3) and (4) properly describe dissipation in dynamic nanoscale interactions and that dissipation during sample deformation can be decoupled from the long-range dissipation, i.e. when mechanical contact does not occur. Several methods to decouple contact and long-range dissipation experimentally and in ambient conditions are proposed at the end of this work.
First assume that the energy dissipated during deformation and at a given separation z ci is E i , where i is a suffix standing for the experimental data point at z ci . Since δ M > 0 is required, experimentally, the data should be acquired in the repulsive regime RR (see below). Then E i reads There are only three unknowns in this equation, i.e. α, η and δ i . For simplicity the maximum deformation for a data point i, i.e. δ Mi , has been written as δ i . Then, taking two arbitrary data points i and j, where µ (µ = −1/F AD ) and ( = 2(2 1/2 )/( R 1/2 ω)) are two known constants. Moreover, K i j is defined as Note that since E and A are also known experimentally, if the sample deformation δ is known, (19) and (20) suffice to quantify α and η. Thus, the last step requires finding δ for, at least, two data points, i.e. δ i and δ j . Use is now made of (6), or (9), to relate δ i and δ j . In terms of deformation δ, where That is, d mi j is the difference in minimum distance of approach for the two data points i and j. As stated, z cji is known to the precision of the z-piezo and z m ji can be obtained experimentally by subtracting the minima from the Fourier series at the two data points i and j. Thus, with (22) and (23) the deformation for two data points i and j can be related exactly.
Although such approach would produce the greatest accuracy and can be readily implemented experimentally, here, for simplicity, we assume that no higher harmonics are excited. Then and the relationship between deformations reduces to a difference between observables Now we are ready to develop a formalism to extract the unknown parameters δ i , α and η from known experimental parameters alone. Note that from (25) a single value of δ i is required to find the deformation at any other point δ j . Thus, there are four possibilities, but one is trivial since it implies that both α and η are zero and reduces to E i = 0 for all i. The other three are (i) α > 0 and η = 0, (ii) α = 0 and η > 0 and (iii) α > 0 and η > 0. For the first case, η = 0 and α > 0. Then, from (20), K i j = 0, and from (19) it follows that for any two data points i and j If the above identity is true, then η = 0 and δ i and α can be obtained as a function of known experimental parameters as For the second case, η > 0 and α = 0. For any three arbitrary data points i, j and k, it follows from (20) that K i j = K ik > 0. Then, from (19) and for α = 0, it follows that for any two data points i and j, Neither the contribution due to F α nor the contribution due to F η is shown. These forces should be responsible for the dissipation occurring during sample deformation δ > 0 (gray colored region). Note that dissipation due to hysteresis already occurs at the longer, non-contact, distances (green area) due to capillary forces F CAP . Adapted from [28].
If the above identity is true, then α = 0 and δ i and η can be obtained as a function of known experimental parameters as The third case is the most general since η > 0 and α > 0, but then three data points, i, j, k, are necessary. The general relationship is where, provided that (i) K i j > 0 and (ii) K i j = E i /A 1/2 i δ 2 i , neither η nor α is zero. Then, from (32) the deformation at a point can be extracted. The deformation at any other point follows from (22) and, finally, α and η are obtained from (19) and (20).
An example of how the above expressions can be used to find δ M , η and α is given next and compared to the results of numerical integration. The data point i = 1 or δ 1 is taken as the reference deformation and is termed δ M . The deformation at any other point δ i can be found from (22). Let us consider the general case for which δ M > 0, η > 0 and α > 0 or K i j = K ik > 0 and i . While any arbitrary model (other than the dissipation during deformation) for the tip-sample force F ts (d) can be used as an example, here the dependence of F ts on d shown in figure 2 is employed. Such a force has already been shown to reproduce the empirical phenomena in ambient AM AFM for several samples and has the advantage to also incorporate a long-range dissipative mechanism, i.e. the capillary force or F CAP . Thus, in the expression employed in figure 2, there are three dissipative mechanisms, i.e. F α , F η and F CAP . The detailed expressions and interpretation of the force distance dependencies in figure 2 can be found in the supplementary data (available at stacks.iop.org/NJP/14/073044/mmedia) and in the literature from which figure has been adapted [28]. The equation of motion is This choice of k allows us to consider the simplified form of d mi j , i.e. d mi j = z ci j + z 0i j − A ij , since the excitation of higher harmonics is then inhibited. The only unresolved step before the above formalism, equations (26)-(32), can be employed relates to the decoupling of the energy dissipated during sample deformation E i from the long-range dissipation. In the numerical example in this work, the bi-stability in AM AFM is used to decouple the long-and short-range dissipation. Bi-stability is a common phenomenon in ambient AFM [21,30]. In particular, for a given critical value of free amplitude, i.e. A 0 = A c , two amplitude branches (low-amplitude branch or attractive regime AR and high-amplitude branch or repulsive regime RR) can be recovered [14,21] (see figure 3 for an experimental example). Assuming that the dissipative processes involved in long-range dissipation are captured in the AR (E ARi ) and that the dissipation involved in sample deformation occurs in the RR only (E RRi ), it follows that for a given data point i or amplitude A i , E i = E RRi − E ARi . Note the two values 9 of ϕ, i.e. ϕ ARi and ϕ RRi , in figure 3 for a given amplitude value A i . E RRi − E ARi can be readily computed experimentally and/or in simulations with the use of either (1) or (2). Note that the gap between the AR and the RR in figure 3, i.e. z c , for a given amplitude A, is approximately 2 nm in the example ( figure 3(b)). The implication is that if the maximum sample deformation δ M in the RR is δ M < z c ≈ 2 nm, no sample deformation should occur in the AR. This condition, i.e. δ M < gap in z c in the AR and the RR, ensures that the dissipation occurring during intermittent mechanical contact is only captured in the RR. Also note that deformations of 1-2 nm or less should be expected in AM AFM for intermediately compliant and/or stiff samples, i.e. E s 1-10 3 GPa, and with the use of standard cantilever characteristics and operational parameters [1,21]. Examples of similar amplitude and phase curves as that shown for aluminum in figure 3, and where the AR and the RR present gaps between 1 and 3 nm, have recently been shown to be readily obtainable experimentally on a variety of samples such as mica, aluminum, quartz, graphite and silicon [28]. Nevertheless, as a rule of thumb, bi-stability might not be present when the sample is very compliant and/or when the stiffness of the cantilever is not large enough [31] (for a thorough theoretical description of the conditions in which bi-stability should be present, see [30]). When bi-stability is not present however, intermittent mechanical contact can still occur provided the free amplitude is large enough [21,29]. Since, if bi-stability is not present, the two amplitude branches are not available, the E i = E RRi − E ARi method presented above cannot be employed. In these cases, several other approaches might be appropriate. For example, since capillary interactions are one of the main mechanisms of dissipation in the long range [3,32], experiments could be performed at low relative humidity and thus all the energy could be assumed to be dissipated during sample deformation in the RR regime. Then the approximation E i = E RRi could be employed. Note that the RR would be the only available regime for sufficiently high values of free amplitude A 0 in these conditions [29,30]. Another possibility consists of using sufficiently large values of free amplitude so that the E i = E RRi approximation is reasonable irrespective of any long-range dissipative mechanism that might be present in the interaction. In short, since the magnitude of the energy dissipated and the sample deformation scale with increasing A 0 [16,17,21], the longrange dissipative processes could be assumed to be overwhelmed by the dissipative processes that prevail during sample deformation. This would particularly be the case provided that A 0 is large enough and when probing very compliant samples, since both the energy dissipated via surface energy hysteresis and viscoelasticity rapidly scale with δ M , i.e. equations (5) and (11)- (12), respectively. In turn, δ M should be relatively large for compliant samples under these conditions. Note that large-enough values of A 0 can be unambiguously defined as values for which A 0 A c [14,21]. This follows from the fact that, even when bi-stability is not present, a critical value of free amplitude A c for which the repulsive regime is reached, can always be defined under standard conditions for a given cantilever-sample system [14,29,30]. The condition A 0 A c indicates that the RR dominates the interaction and sample deformation occurs. Again, for very compliant samples this guarantees that sample deformation is of the order of nm. Finally, the E i = E RRi approximation would be reasonable in these conditions. As a final note in this topic, it should be noted that the question of whether long-range dissipative interactions are prevalent in the repulsive regime, together with the short-range or contact dissipative mechanisms discussed in this work, could be probed with the dE dis /dA method [15]. Thus, since in the absence of bi-stability the E i = E RRi − E ARi method cannot be applied, if the dE dis /dA method shows that long-range interactions are large relative to the contact processes, the E i = E RRi approximation might lead to large errors in the present formalism and ingenuity in the experimental approach would be required. In particular, note that the dE dis /dA method shows how to identify contributions from interfacial long-range interaction which might be prevalent in several scenarios in AM AFM [1].
In the simulations of the example presented here, the condition E i = E RRi − E ARi can be employed since bi-stability is present. In particular, the critical value of A 0 in the example is A 0 = A c ≈ 9 nm (curve not shown). Moreover, the gap between force regimes, i.e. z c (see figure 3(b) for an example), is of approximately 1-2 nm. As stated, if the maximum sample deformation δ M is less than this gap, i.e. if δ M < 1-2 nm, both amplitude branches can be recovered with sample deformation only occurring in the RR as required. In the example in the simulation, this condition is satisfied since the maximum sample deformation is less than 1 nm (see below) agreeing with previous studies for E s 1-10 3 GPa [1]. Although only three data points are necessary to estimate the values of δ M , η and α, several data points are used here in order to establish the accuracy of the technique numerically. Furthermore, this does not restrict the practicality of the method since, in experimental curves, a very large number of data points can be collected (see figure 3). In the numerical simulations, N = 21 data points have been used. This leads to 1 + N (N − 3)/2 = 190 estimations for δ M (32). Taking as the reference the unknown deformation δ M at z c = 5.4 nm, the value δ M = 0.6717 nm (mean value) is recovered at z c = 5.4 nm with the use of (32). Note that, as stated, prior knowledge of the absolute value of z c is not required to find δ M . The standard deviation is ≈8 pm. The true value of δ M at z c = 5.4 nm, according to numerical integration, is 0.6718 nm. The error is thus a fraction of a pm. This demonstrates the potential accuracy of the method. Note that the Chauvenet rule has been applied to the 190 values of K and only two values have been excluded as outliers. Errors are due to the assumptions in the derivation of E η (12), the use of the simplified form of d mi j (24) and the lack of accuracy in the numerical integration of the equation of motion (a step size of 4096 data points per cycle and a standard Runge-Kutta algorithm of the fourth order have been used in the numerical integration). Once δ M is known, η and α can be recovered from (19) and (20). N (N − 1)/2 = 210 estimations for η and α can be computed. The recovered (mean) value for η is 503 Pa s with a standard deviation of 41 Pa s and where a single outlier has been excluded from the data set (32 estimations have been excluded from a physical point of view since these were negative). Recall the true value is η = 500 Pa s. For α the recovered (mean) value is 0.505 with a standard deviation of 1 (two outliers and 33 negative values excluded). Recall that the true value is α = 0.5. Finally, we note that three solutions for K i j − K ik = 0 are typically found for any three given data points i, j and k. Only one solution produces a real positive deformation δ M , where K i j > 0. This work shows that by monitoring the magnitude of the energy dissipated in the tip-sample interaction with sufficient accuracy and by correctly establishing the dissipative mechanisms during sample deformation, the tip-sample deformation can be recovered with, in principle, pm resolution. It has also been shown that fundamental dissipative properties of the sample can also be recovered quantitatively. In particular, viscoelasticity and surface energy hysteresis can be quantified. Experimentally, errors in the acquisition of amplitude A (parts per thousand [19,33]) and the phase ϕ (fractions of degrees [4]) should be expected to affect the accuracy of the method since the formalism developed here (equations (26)-(32)) depends on the accuracy in the measurements of these observables. As stated elsewhere [19], this is a general limitation of AM AFM methods. Other limitations include the calibration of other otherwise known parameters such as the tip radius and the stiffness of the cantilever but this is a field of research where developments are being made continuously [12,14,34]. More fundamentally, deeper insights into other dissipative channels such as the capillary interaction are being reported [18,28,35,36] and these might imply that future developments of the theory presented in this work will potentially increase the accuracy of the recovered parameters. As a final note, it should be noted that dissipative properties might depend on nanoscale dimensions. In this respect, developments regarding the characterization and accurate calibration of the tip radius in situ [14] might be valuable in future experimental work and in relation to the present formalism. That is, by probing the sample with a range of tip radii, the dependence or otherwise independence of dissipative properties on dimensions could be probed. This could have implications on a theory where the interface between atomic and macroscale phenomena is studied. In summary, this work establishes a robust framework to quantify dissipative properties with true nanoscale resolution in ambient conditions.